r-

Characterization of Brick Masonry Stiffness by Numerical Modelling and in si tu Flat-Jack Test Results

L. JURINA *, A.PEANO **

(*) Dept. of Structural Engineering, Politecnico , Milano, Italy

(**) Istituto Sperimentale Modelli e Strutture (ISMES), Bergamo, Italy

Abstract- A large seale, non destructi ve, in-si tu test for determining the mechanieal properties and the stress state in masonry walls has reeently been developed at ISMES. The teehnique is based on inserting flat-jaeks between the briek eourses. The relative displaeements of various points loeated on the wall surfaee elose to the flat-jaeks are measured as the pressure is inereased.

In the paper a numerieal simulation proeedure is presented. The purpose is to determine the material property values which minimize the diserepancies between eomputed and measured displaeements. Masonry is modelled as an elastie orthotropie material. In order to reduee the number of indepen­dent unknowns in the eharaeterization proeedure a method of relating the overall orthotropie stiffness eoefficients of masonry to the properties of bricks and mortar is developed.

1. INTRODUCTION

Struetural analysis of masonry buildings requires masonry to be modelled as a homogeneous material.

It is however quite diffieult to obtain meehanical properties for actual masonry. In principle it would be suffieient to take an undisturbed sample of masonry, of adequate size, and apply standard laboratory tests, disregarding the intrinsie nonhomogeneity of the material . In practiee such an obvious pro­eedure eannot always be applied. Typieally it is usually impraetieal to obtain large samples of the masonry of existing buildings (beeause of the high eost, of the structural risk or of the respeet due to aneient historie monuments).

The problem has motivated the development and sueeessful applieation of a new non destrueti ve in-si tu large seale testing teehnique . The new testing proeedure is based on the applieation of two flat jaeks, inserted between bri­ek eourses. The masonry deformations under known pressure loadings are measur­ed in order to derive all needed information on the meehanieal properties of masonry. Sinee two mortar layers only are destroyed, the proeedure may be ap-

177

178

plied at all points of interest to the structural engineer wi thout the res­

traints due to aesthetic, cultural, economical or technical considerations.

In the paper the testing procedure is sketched very briefly. A comp lete

treatment is available elsewhere [lJ . The paper is centred on a computer pro­

cedure to be used for interpretation of the experimental results.

The interpretation is based on the comparison between in-si tu deforma­

tions and those computed by a three dimensional finite element model of a ma­

sonry portion around the flat-jacks. In the computations the masonry is mo­

delled as an equivalent orthotropic elastic continuum.

There are two reasons for anisotropic behaviour of brick masonry. The

first one is simply the texture of the two component materials. As brick and

mortar layers are located at regular intervals along three mutually orthogonal

directions, an orthotropic material model seems very appropriate for large

scale behaviour of masonry. The second reason is that bricks often exhibit a

markedly orthotropic behaviour, due to the extrusion and lamination processes.

In the following anyway both mortar and bricks are considered isotropic. In

fact we have in view the application of the proposed technique to old

buildings, where molded bricks are used.

Orthotropic materials are characterized by 9 independent elastic coeffi­

cients. The identification of such a large number of unknowns is computational­

ly expensive and error prone because of the limi ted amount of experimental

data generated by a single testo On the other hand the assumption that brick

masonry behaves isotropically may be unnecessarily crude. The paper presents

an analytical procedure which yields the 9 coefficients of orthotropic elasti­

city as a function of the material properties of bricks and mortar and of the

thickness ratios of mortar and brick layers in the three principal directions

of orthotropy.

Since for a given masonry wall the thickness ratios can be very easily

measured, the parametric search depends only on the elastic moduli and the

Poisson's ratios of mortar and bricks. In practice considerable changes in the

value of the Poisson' s ratio have a limi ted effect on the resul ts, therefore

the elastic moduli of mortar and bricks may be assumed as the primary unknowns

to be identified from the experimental data. Since the interaction between

mortar and bricks is too complex to be taken into account exactly, the moduli

determined from the experimental data should be considered only as "equi­

valent" moduli which are likely to be influenced by the thickness ratios

between mortar layers and bricks. Once the equi valent moduli of mortar and

bricks are determined, the above mentioned analytical procedure yields the

desired equi valent orthotropic material stiffness of masonry. It is expected

that the error in the equi valent masonry stiffness be much smaller than the

error in the elastic moduli of mortar and bricks.

2. TESTING PROCEDURE

The in-si tu testing procedure is comprised of two stages: a) deter­

mination of the state of stress in the masonryj b) determination of strength and stiffness characteristics.

In the first stage of the test a horizontal cut (wide 40 cm, deep 20 cm

and high 1,5 cm) is made in the masonry wall by removing a mortar layer

between two bricks courses.The consequent stress redistribution in the masonry

determines a partial closure of the cut , which is moni tored by measuring the

re I ative displacements of previously established reference points on the wall

surface .Then a flat-jack is inserted in the cut and a n increasing pressure is

appl ied until the refe rence points go back to their ini tial location. This

procedure provides the local average vertical stress. Occas ional discrepanc ies

i n the reference points locations are caused by local collapse of the masonry

during unloading or to difficul ty of restoring exactly the ini tial distribu­

tion of vertical pressures.

In the second stage another flat-jack is inserted about 50 cm above or

under the first one. The two jacks, connected in paraI leI to an hydraulic pump, undergo several loading/unloading cycles. At regular load intervals, the relative displacements of alI reference points are measured.

AI though the flat-jacks pressure is not increased up to the leveI of

masonry collapse, the strength of the masonry can be estimated by extrapola­

tion from the load deformation diagrams.

The stiffness of the masonry can be determined in two ways. First a

physical model, wi th known material properties, may be used to calibrate the

te~ting procedure. This approach is illustrated by Rossi in a companion paper

presented to this conference [lJ . The second approach, namely interpretation

by computer simulation, is dealt with here.

The above mentioned testing procedure has been devised and first applied

in view of the static restoration of the Palazzo della Ragione, an old masonry

building in Milan [2] . The mechanical properties o f the masonry of externaI

walls was needed for fini te element analysis of the deformations caused by

differential foundation settlements.

fLAT JACK

flAT JACKS

FIG.1 - F1at-jack test: scheme of the 2nd stage

179

3. EQUIVALENT MASONRY STIFFNESS MODEL

The "micro-scale" state of stress of masonry under vertical loads is

three dimensional because of the different deformabili ty of the component materials [3]. The lateral deformation of mortar is usually restrained by shearing forces applied by the adjacent stiffer bricks. Therefore mortar is in

compression in all the principal directions, while bricks are subjected to vertical compression and horizontal tractions. At a finer level of detail the

state of stress is very complex because the transfer of shearing forc es from bricks to mortar determines a complicated stress state which depends on the

relative stiffness of the two materials and on the geometric pattern of

masonry. Hence it is very difficult, if at all possible, to r elate the exact material properties of the component materials to the exact "large scale"

stiffness of masonry [4] The aim here is different because the exact value of the elastic modulus of mortar and bricks is not seeked. The aim is

identification of large scale properties of masonry and i t is desired to

exclude from the parametric search the possible orthotropic material stiffness

matrices which are not compatible wi th the

known micros tructure of

masonry. In fac t mason­

ry is a special ortho­tropic material compri­

sed of only two isotro-

pic components arranged at regular and known

geometric intervals.

Therefore the 9

stiffness coefficients of orthotropic elastici­

ty will be related to other 9 coefficients: 3

parameters determinect

by the geometric pat­tern of masonry and mor­

tar texture, and 3 ma­terial constants for

each of the two compo­nent materials. The

three geometric parame­ters are the three ra­tios between the brick dimensions a. , i

1,2,3 and the ~orrespon­ding thicknesses of mor­tar layers t. , i 1,2,3. The mateiial con­

stants are obviously the Young' s moduli (E), the Poisson's ratios

180

~--t----~----1--'" ;1iii L..--t----+--- _J

a) b)

FIG.2 - Deformation of brick masonry in compression

and in shear: a) Real masonry texture;

b) Fictitious masonry texture.

( v) anel the shear moduli (G) of mortar and bricks. In this paper only the

Young's rnoduli are varied beCalJSe the Poisson's ratios have a limited impact

on t h e final results and tbeir volue is 3SSU~.led a-priori. Moreover the shear

modul i are dependent variabl es because oí t!1e isotr opy of the component p.' é'.te·­

rials and are computed as:

G E

In conclusion the Young's moduli of mortar and bricks are the on l y independent

va r iables to be identified in the followi' lg. As discussed earJi er t:hey should

be viewed as equivalent rnoduli. In order to detel'mi'lA the equivalent masonry

rigidity it is necessary to isolate a repetitive module iIl the masonry

pattern . Here we c hoose a single brick supplemented by layers of mor tar on

three orthogünal faces. This module is çons:iderabl y ,;'are complex than the ones

used by other authors [5,6,7J . Still such a simple b1Jilding ",odule does not

distinguish between the real mo.sonry p,tte'n (fi8.2a) 2.nd the fi.cti tious one

indicated in fig. 2b. On the other hand the two geometJ 'ic róltterns 1 ead to the

same large scale behav ioUl', as long a s ::lX ial s t iffness i s c ons idered. Of

cüurse a discrepancy rnay occur for the she,'1ring stifrncss becaus e re,ü masonry

behaves more like a horizontally stratificd ma ter ial . In fac t shear il1g defol ma­

tion is mainly due to the mortal' layers a, .. d the mOl tar ':letween bricks is

restrained by the higher shearing rigidi ty oí' the br ·Lc !?:s. The equ"i va l ent shear­

ing stiffness for a strati fied rnateria) is alre:;tdy ava ' lahle in the technical

li terélture [ 8J . T\1e shearing stiffness providpd by the ',·odel proposed here i s

a l ower- bOl:nd. The real stiffness is cer tai.nly higher ' , but stiJl lower than

the vah!p. provicted by a strat i.fi.er' '1ate! i al modelo As i ndi cated i n the

e xp l o ded view i f.i_g . 3) the referel1ce 010(luli can be viewe d as compl'ised o f 8

blocks . One of t h e blocks i s ehe br j.rck. the cth e r' are portions of the mortal'

12~ye<! ' s det€lf"L-;!'~ '3 d cy the p 13Il~S of ~,'~e b.':"' .i.r~ L f~v:' es i.r!si.~.e t~1 S ;:ererpn:; e mo,jl~l e.

D QCOt"'lr::sjti o11 (} tl.LE" 1 2. p P':j ~oi..r.. r~ (.,'; :J.. f' i1,"1.) '_'1'18 to'>.ll.l" (b,. 9 :J :3 "/ ,rol "tp r"

bl-'..: ~-ks .

181

182

The fol lowing assumptions are made:

1. Each block is in a state of constant equi valent strain. The strain is called "equi valent" because the actual strain and stress state is more

complex, as discussed earlier.

2 . Compatibility of equivalent strains of the various blocks is enforced

exactly.

3. The equi va lent strain of each block is related to a constant state of

average stress by the above-mentioned equi valent properties of brick and

mortar.

4. Normal and shearing forces applied to the module are in equilibrium with

the resultant of the internal forces supplied by the blocks.

5. Equilibrium of resul tant normal and shearing forces is also enforced

across the three internal planes which generate the internal subdivision in 8 blocks.

The above procedure may be applied independently to shear stresses and

strains and to normal stresses and strains. In fact the two sets of equations

are uncoupled. This is due to the orthotropy of the equivalent material stiffness matrix. The equations corresponding to the above assumptions are not given in detail here [9].

3.1 Normal stiffness of masonry

Let us first consider normal strains and stresses. There are 24 constitu­

tive equations, 18 compatibility equations and 6 equilibrium equations. The

unknowns are the 24 normal stress components (three for each block) and the

corresponding 24 normal strains. Compatibili ty equations for elongations in

the i-th direction simply state that the four blocks located at the same distance from the j-k plane have the same elongation (three equations). Six

such equations may be written for each direction, as two groups of four blocks

each are generated by each of the above-mentioned internal planes.

The deformation of the repeti ti ve module is governed by the 6 axial

strains of the brick (b) and of the little mortar block (m). The axial strains

of the other 6 mortar blocks are eliminated by means of the compatibility equations .

Therefore the average strain of the module is computed by means of the strains of the brick (b) and of the mortar block (m):

b ti

m + a. E i + E i

E. = ,

(2). ,

ai + t. , The six unknown strain components can be determined by means of the equili­brium equations. For instance the two equilibrium equations for the i-th

,

't-

d irection are:

+ b b ' bk mi ( 3a) ai .Q,j \ ai a. a k + a. J t. a k + ai a. t k + ai t. t k J 1 J J J

+ bi mk t j mj t k

m t j t k (3b) a. .Q,j Q,k = a. a. a k + ai a k + a . a j + ai 1 1 J 1

Where 1 = a + t and 1 = a + t are lengths of the sides of the module and s upersc)ipts j deno~e k k k

the block according to the notation of fig.3. The six

e quilibrium equations are in general coupled. The coupling is caused by a

nonzero Poisson's ratio in the stress/strain relationship for mortar and

brick. In general the analytic treatment of the problem is quite complex and

i s omi tted here for brevi ty [9]. Al thought in the following we assume

v = 0.15, the principal elastic moduli for the case v = O are presented here.

They can be derived rather easily from the above equations:

(4) Q,i .Q, j .Q,k + (p - 1 ) Q,i a 1 a k

.Q,i .Q,j \. + (p - 1) t i a j a k

Where P = E /E is the ratio of Young's moduli of brick and mortar. Equation b m

(4) indicates that the anisotropy is more pronounced for high values of the

br ick Young' s modulus, as expected. Moreover there is an upper limi t to the

br ick masonry moduli determined by the thickness ratios:

(5 ) t.

1

Analogous properties are exhibi ted by the moduli for Poisson' s ratios larger

t han zero.

3 .2 Shearing stiffness of masonry

Again there are 48 unknowns: the 3 shear stresses and the 3 shearing

strains of the 8 blocks. The problem is simpler, however, because no coupling

be tween different planes exists.

Let us consider shear deformations

in the i-j plane (Fig.4). The first compatibili ty equation enforces conti­

nui ty of deformed blocks at the inter­

na l point where all blocks meet. This

requires the sum of the angular deforma­

tions to be zero:

(6) mk bj - O + Yij - Yij -

The deformation of the mortar layer of

th ickness t is completely governed by k

the deformation of the other larger

•• I

FIG.4 - Equivalent shear deformation

of the module

183

184

blocks. This implies 4 additional compati.bility equations. Four addi tional equations resul t from enforcing equil~bl'i.um on the exter .. ,

nal surfaces of the repeti tive module . A typical equation is:

(7)

Note the simiJ ari ty wi th the traslational eQilÍl ib.cium eqnations. Here . hOIll'­

ever , only three equilibr i' lm are independent , because of the overall rotatio­nal equilibrium of the module .

The solution of the above equation can be marketl. out expl ~Cj. tly. In order

to deduce the large scale shearing stiffness of masonry i t i s also necessary to define the average shear strain yt of the module . The snear strain is the

sum of the two angles 8jj and 8ji indicated in fig . 4 . After some compllta-tion it t:urns out:

(8) + b bi bj t j mk

t j ) I ti "( j j = (y; j a . a j + r ij ti Qj -t Yij ai lo Yij ti R-j 1

From the above results the a verage shearing modulus o f masonry is:

.;- i R-j -\ + (0'-1) (ai a j + ~-~. ã j ti) aI{ (9) Gij = Gm

R-i tj -\ + (a - l)(a i t j + a j ti) ak

where CT is the ratio between the shear modulus of the br ick and tne S,ledr

modulus of mor tal' . Equation (9) is a generalization of the formulas provided by Pinto for

stratified materials. In fact assuming t . = t . = O OI' t . = t . = O we obtain respectivelJ the shear modulus in the plkJe orl the stratã 01' ihe shear modulus in a plane normal to the strata .

As discussed earlier the above madel is likely to understimate the shear modulus of masonry . Let the axis 1 to be horizontal i n the plane of the wall ,

the axis 2 be horizontal normal to the wall and the axis 3 be the vertical one. The geometric pattern of masonry in the plane 2-3 is similar to the one

indicated in fig.2b . Therefore equation (9) can be applied. In the other two planes the texture is as indicated in f j g . 2a . Hence we resort to a different

model o Let u s assume for the moment a str atified continuum composed of

al ternate layers of shear moduli G an d G and thickness a and t. The b m'

equi valent shear modulus (G*) in a pla."1e nor~al to the strata i s such that :

(lOi

a + t ---c;+- = +

In pra~ti e a brick COUi~se is not a continuum strat:um because o f the mortar between adjacent bricks , therefore its stiffness must be reduced accordingly .

After some labor , i t is found that the shear moduh in planes 1-3 and 2-3 should be computed as :

" .i, ~, j + ( O - 1 ) ti a.

(11) li: . = .... :::L· l J fll ·.2

i Rj + (C1 ~ 1 ) ti aj

4. NUMERICAk PROCEDURE _~~HARA~TER~~ON OF MECHANICAL PROP~~IES O~ MASONRY

The non-destructive flat-j aek test was developed and first applied to the

meehanieal eharaeterization of t 'he briek masonry ~Jalls of Pala:!:zo della

Hagione, a monumental building in Milan [2J. The bui ldi.ng is eompri sed of a lower part of XIII eentury wi th marked

craeks due to differential foundation settlements and of an upper part of

XVIII eentury . where poorer masonry materiaIs were used. During the las t two

centuries up to 1959 the building hosted the Notarial Archives af the town. The hea.vy load aecelerated the deterioratioll of the building.

In view of the statie restoration, an extensive testing program was con­ducted. The pr ogram ineluded 6 flat- Jack tests on the masonry. An aceurate

interpretation of test resul ta can be based 011 the eharaeterization procedure descr ibed in th", f0110wing. Characterizati on OI' identifieation problema are well known [10J. Struetural engineering appl ications ~re re1atively new, however. See for instanee [11, 12J .

Two k ey ingredients are needed : a parametrie numerical model and an error fUl.\ction· to be used to fle1eet the values of the paz'ameters that, when adopted in the numerieal model, lead t o computed values "as elese as possible" to the

experimental ones. Here a 3- D fini te eJement mode 1 of a portion of masonry surrounding the

flat- ja.eks was prepared. Beeause of simmetry only one quartel' of the problem

was modelled using 18 60 degrees of freedcm. The model covered an aY 'ea of

1, 50x1 ,50 square meters . As the Poisson I s ratio of botl! mortal' and br.teKs was assl.lílled 0,15 [5] , two parameter s only (E and E

b) r!9d to be deter!"l5. n.ed .

The er-ror f'tmction can he se Tec ted ~:: mp]y as :

(12) z

wL'. e r e:

c l:.(u.(E't.,. E ) -1 1 _ m'

um is the vector of measlJred displacements j!J the in-s i tu test

c is the vector of" displacelTlents çal~ulated in the model u

On toe basis of the assumption of linear elastie behaviour, it can be observed that if vector Oe represents the calculated displaeements corresponding to a

r-2-.-2- c pair of values (E, E) sueh that V Eh +E '" 1 , fi /0'>- represents the calculated disp laceJ/ents b corresponding t o - the m elastic modul i cI.. E and ()(. Eb' Henee th€' err or function can be expr essed in the t 'ollowing we.y : m

(13 ) z whe re, as earlier, P ~ E / E .

For a given ratio pb p ~he local mÍl'ümum Of' the quadr'atic e r'l'or f unetion Z in the domain (E , E ) can be f ound ve ry t'asj Iy obsE't v ina; tloe.t:

b 10

185

(14) CI. = ~c - 2

L;{U;{p))

The search for the absolute minimum value of Z can be performed in a param e­tric way by choosing a sui table number of ratios p , along which a local

minimum is calculated . Due to the different order of magnitude of vertical and horizontal di­

splacements, two indipendent minimizations where performed. After averaging measurements simmetric wi th respect to center lines, 4

independent vertical displacements and 4 independent horizontal displacements were obtained for each position in which the flat-jack test was performed. If one tries and applies the characterization procedure using the 4 vertical displacements only, a case of non identifiability arises.

A 'locus' is obtained in the (E , E ) domain characterized by almost the same minimum value of the error functPon fline A of fig.5).

This is not surprising as the same average vertical modulus can resul t

from different combinations of elastic moduli of the component materiaIs.

Using as measured displacements the horizontal ones only, a second '10-cus' of minima is obtained (line B of fig.5). The intersection of lines A and

B corresponds to the pair of moduli E and E minimizing the error functions of both vertical and horizontal displ~cement~. It is interesting to observe the trend of the coefficients of the elastici ty matrix corresponding to the points marked on line A and line B. It can be noted from Table I that, even

though Eb and E are not separately identifiable, the value of the average vertical stiffne:s E is practically constant for alI the points of line A.

The same consid~rations can be extended to the average horizontal stiff­ness E

1 corresponding to

the points of 1 ine B. The

intersection of lines A and B provides the best fit­

ting of alI measured defor-mations.

5. CONCLUSIONS

The theoretical model for sti'ffness of brick ma-

.fOOO

3'000

2'000 sonry developed in this pa- p -IO

per is very effective for identification problems, 1'000

where the number of unknown parameters must be kept to a bare minimum .

So far no attempt was made to correlate the com-puted equi valent modul i E

and E with the rea~ Young' s m modul i of bri cks

186

o o 5'000 10'000 15'000 Eb

FIG.5 - Parametric identification of equivalent mortar and brick moduli from vertical (A ) and horizontal (B) deformations.

....... ----------------TABLE I - Material constants appearing in the orthotropy matrix for line A

E /E E1

E2

E G G G V12 V23 V31 b m 3 12 23 31 (MPa) (MPa) (MPa) (MPa ) (MPa) (MPa)

1 4167 4167 4167 1811 1811 1811 0,150 0,150 0,150 3 4694 4464 4237 2344 1710 1942 0,144 0,139 0,130 5 5376 4784 4291 2721 1623 1946 0,135 0,125 0,109

10 6711 5291 4347 3351 1510 1920 0,118 0,103 0 , 078 15 7692 5586 4385 3747 1455 1899 0,105 0,091 0,062 20 8403 5780 4397 4021 1423 1884 0,095 0,083 0,053

and mortar. The computed values are likely to overestimate the real ones as in the theoretical model local equilibrium is violated. Shear stresses cause also local axial strains and axial stresses cause also local shear strains, particularly at the interface between different materiaIs.

A possible source of errors in the characterization procedure is the pre­sence of creep effects which may develop during the experimental tests [6J.

As a general recommendation for good characterization results it is suggested that the portion of masonry to be tested and the number of displacement measurements should be as large as possible. Inclined displace­ment measurements (to be interpreted separately) would yield a third indepen­dent locus. This could allow a better definition of the intersection point and an indication of the scatter of the identified parameters.

An extensi ve testing programme using well known component materiaIs has to be developed to assess the validi ty of the numerical characterization procedures here described and to evaluate the confidence limits of the results .

ACKNOLEDGEMENTS

This work was partially supported by the National Research Council (CNR) of Italy.

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P. P . ROSSI: Analysis of mechanical characteristics of brick masonry by means of non-distructi ve I in-si tu I test - Proceedings 6th IBMAC, Roma I

1982

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[3J H.K . HILSDORF,: Investigation into the failure mechanism of brick mason-

187

ry loaded in axi aI compl~es3ion - l'roceedings lnt . Conr . on _ Masonry Structural Sys cems, Texas, 1969 - Gulf Publishing Co., Houston, Texas ,

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effect on masonry behaviollr - Proceedings Ira Canadian Maso1ll7 SYlT'po-, ~, Ottawa, 1980

[6J E.L.JESSOP, N. G. SHRIVE , G. L " ENGLAND: Elastic and c r eep propel ties a f masonry - Pi'oceeaings Nor th Ameri can Masonry Conference, Boulder , ColcI'a­do, 1978

[7J N.G . SHRIVE, E.L .JESSOP , M. R. KHALI1 : Stres s - straln behav~our of masonry

walls, - Proceedings 5th IBMAC, Washington D.C " 1979

[8J J . L. PINTO: Stress and straina in an anisotropic - orthotropic body -Proceedings 1st Cong. of l.S . R. M. , Lisbon, 1966

[9] A. PEANO, L .JURINA,: A theoreti cal model for de formational behav iouI' of

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[10] P . EYKHOFF: System ldentificatiún - John Wiley and Sons, London , 1974 .

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Engineeri,,!!g , n . 4/1974

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Formations, Capri, 1977

188